# What is known about these spherical point distributions related to pinecones?

It is well known (e.g. Vogel, "A better way to construct the sunflower head", 1979) that the pattern of florets on a sunflower head can be recreated by drawing points at polar co-ordinates $(\sqrt k, 2\pi k\phi)$, as $k$ ranges over integer or half-integer values $\ge0$; here $\phi=\frac{\sqrt5-1}{2}$ is the golden angle, and the radius of $\sqrt k$ ensures that the first $n$ points lie within a disc of area proportional to $n$.

It is also well known that on a unit sphere, a cap of height $h$ has area $2\pi h$. So an analogous pattern of points on a sphere could be chosen with cylindrical polar co-ordinates $(\sqrt{1-h^2},2\pi k\phi,h)$, with $h=\frac{2k}{n}-1$, as $k$ ranges over integer or half-integer values $\in [0,n]$. (Sorry if the notation or exposition are uglier than they might be.)

These point distributions, and the corresponding ("pinecone-like") polyhedra obtained by taking them as tangency points for faces, seem to me such an obvious candidate for study - both for theoretical interest and applications - that others must have considered them, but so far I've had difficulty finding appropriate search terms. Does anyone know what's been published (or informally noted) on this subject?

For anyone who might be unfamiliar, here's a picture of the planar distribution, from Wikipedia:

• I've often seen this in reference to the subject of phyllotaxis: en.wikipedia.org/wiki/Phyllotaxis though perhaps that's too general. Vogel's article has 200 citations according to Google Scholar. Did you try looking at some of those citing books and articles? – j.c. Jun 16 '17 at 3:10
• The citation list has a lot of things which almost certainly aren't relevant, many things which probably aren't, and so far none which probably are. You gave me the idea to try searching for the title of Vogel's paper alongside the word "spherical", which as far as I can see throws up a bunch of people rediscovering the same concept outlined here and going over basic properties. I haven't yet found anything particularly interesting, but will update if I do. The latter search also suggests the keywords "spherical Fibonacci", though the number $n$ of points need not itself be a Fibonacci number. – Robin Saunders Jun 16 '17 at 3:19
• I note with some despair that more than one of the aforementioned rediscoverers appears to have successfully obtained a patent for the result. – Robin Saunders Jun 16 '17 at 3:21
• The paper "Spherical Fibonacci mapping" by Keinert, Innmann, Sänger, and Stamminger cites most of the other papers on the subject that I've found so far; it also contains an overview, a new practical result and a few nice pictures. The main paper not cited is "Point Sets on the Sphere $\mathbb S^2$ with Small Spherical Cap Discrepancy" by Aistleitner, Brauchart, and Dick, which is also the only paper published in a journal that's decisively about maths, and the only one on arXiv (arxiv.org/abs/1109.3265) - the rest are very implementation- and/or application- oriented. – Robin Saunders Jun 16 '17 at 4:26

It is better to look at this points as points on a torus. In this case this set will be Korobov's net (aka good lattice points set). Such sets have least possible discrepancy. For example we may start with $34\times 34$ square (torus) and vector $(5,3)$ (such that $5^2+3^2=34$). In this case Korobov net is the set $$\{(5k\bmod 34,3k\bmod 34):0\le k<34\}.$$ It discribes the structure of the pine cone with $3$ spirals in one direction and $5$ spirals in another direction: